\documentclass[11pt,letterpaper]{article} \usepackage[T1]{fontenc} \usepackage[utf8]{inputenc} \usepackage{lmodern} \usepackage[margin=1in]{geometry} \usepackage{amsmath,amssymb,amsthm,mathtools} \usepackage{mathrsfs} \usepackage{array,booktabs,tabularx} \usepackage{enumitem} \usepackage{microtype} \usepackage[hidelinks]{hyperref} \setlength{\emergencystretch}{3em} \setlength{\parskip}{0.35em} \setlength{\parindent}{1.2em} \newtheorem{theorem}{Theorem}[section] \newtheorem{proposition}[theorem]{Proposition} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \theoremstyle{definition} \newtheorem{definition}[theorem]{Definition} \newtheorem{example}[theorem]{Example} \theoremstyle{remark} \newtheorem{remark}[theorem]{Remark} \newcommand{\D}{\mathcal D} \newcommand{\Dp}{\mathcal D'} \newcommand{\WF}{\operatorname{WF}} \newcommand{\supp}{\operatorname{supp}} \newcommand{\Hom}{\operatorname{Hom}} \newcommand{\im}{\operatorname{im}} \newcommand{\Ann}{\operatorname{Ann}} \newcommand{\Rec}{\mathsf{Rec}} \newcommand{\Recp}{\Rec^{\circ}} \newcommand{\Mmap}{\mathsf M} \newcommand{\Pint}{\mathscr P_{\mathrm{int}}} \newcommand{\Kint}{\mathscr K_{\mathrm{int}}} \newcommand{\Kret}{K_{\mathrm{ret}}} \newcommand{\Ksup}{K_{\mathrm{supp}}} \newcommand{\Ucan}{U_{\mathrm{can}}} \newcommand{\Cross}{\mathscr C_{\mathrm{cross}}} \newcommand{\Rquant}{R_1^{(1)}} \newcommand{\Eout}{\mathscr E_H} \newcommand{\Ejet}{\mathsf E} \newcommand{\Jjet}{\mathsf J} \newcommand{\Ob}{\operatorname{Ob}} \newcommand{\eps}{\varepsilon} \newcommand{\dd}{\,\mathrm d} \newcommand{\PV}{\operatorname{PV}} \newcommand{\Pjet}{\mathscr P_r^{\mathrm{jet}}} \newcommand{\Fz}{F_0} \newcommand{\Sresp}{S_{\mathrm{resp}}} \newcommand{\Natt}{\mathcal N_{\mathrm{att}}} \newcommand{\dist}{\operatorname{dist}} \title{Landmark XV-E\\ Source Recovery for the Frozen Joint Graph,\\ Descending Extensions in the Punctured Normal-Jet Module,\\ and the Boundary of the Full Lift} \author{} \date{} \usepackage{tikz} \begin{document} \maketitle \begin{abstract} XV-E continues XV-D at its gate D0. Its first task is a source recovery pass: to obtain, from the supplied record, the definitions that XV-D found missing for the frozen six-branch joint target, and, if they are obtained, to define the preliminary full lift, evaluate its reconstruction-null defect, and test cancellation in the inherited correction class. The recovery pass finds one change. Excerpts of XII-A and XII-D now in the record display tensor and kernel formulas that are candidates for the undefined symbols $\mathsf A$, $L$, and $R$ in the internal kernels. The record does not identify them with those symbols, and XIV-H's defining equation for $L$ is still not reproduced, so they are recorded as candidates and not used. All other items remain unavailable. No lift for the actual graph is definable, and no graph-specific defect, obstruction, or estimate is asserted. The graph-level outcome is \emph{Unresolved at E0}. The new mathematics concerns the one source-complete module, the finite normal-jet module with punctured reconstruction. We prove that the null defect of an extension lift is governed by the annihilator ideals of the punctured and extended coefficients, with $H/H_T\cong(\Ann T^\circ/\Ann T)^{M_r}$; that null relations divide into universal, flat-annihilator, and jet-detected relations, of which only the last produce a defect, and that the defect is locally a functional of a finite jet along the puncture. For a point puncture we prove that a descending extension exists whenever any extension exists, with an explicit correction by finitely many derivatives of the delta at the puncture; in one dimension every extension descends. For a general puncture we prove that a descending extension exists if and only if a stated relative estimate holds, and we leave open whether that estimate can fail. Finally we correct an overstatement in XV-D's coefficient proposition and show by examples exactly which implications between extendability of a trace and of its smooth multiple hold. None of these results is transferred to the inherited graph. \end{abstract} \begin{figure}[ht] \centering \begin{tikzpicture}[x=1cm,y=1cm] \node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (a) at (0,0) {Source-recovery pass}; \node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (b) at (5,0) {Four candidate symbols}; \node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (c) at (10,0) {A supplied identification}; \node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5,-2.35) {None of the candidates is supplied. The graph stays unresolved.}; \draw[->] (a) -- (b); \draw[->] (b) -- (c); \draw[->] (b) -- (s); \end{tikzpicture} \caption{Candidates are not identifications. The three boxes name objects this manuscript keeps separate. The dashed box is the stated stop. Not a proof, and not an admission of a quantum coefficient.} \label{fig:scope-XV-E} \end{figure} \begin{figure}[ht] \centering \begin{tikzpicture}[x=1cm,y=1cm] \node[draw,rounded corners,align=center,text width=4.3cm,font=\small,inner sep=3pt] (a) at (0,0) {Source-recovery pass}; \node[draw,rounded corners,align=center,text width=4.3cm,font=\small,inner sep=3pt] (b) at (6.4,0) {A supplied identification}; \node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (3.2,-2.35) {None of the candidates is supplied. The graph stays unresolved.}; \draw[->] (a) -- (s); \draw[->] (b) -- (s); \end{tikzpicture} \caption{Candidates are not identifications. The boxes name objects this manuscript keeps separate. The dashed box is the stated stop. Not a proof, and not an admission of a quantum coefficient.} \label{fig:split-XV-E} \end{figure} \begin{figure}[ht] \centering \begin{tikzpicture}[x=1cm,y=1cm] \node[draw,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (a) at (0,0) {Source-recovery pass}; \node[draw,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (b) at (0,-1.7) {Four candidate symbols}; \node[draw,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (c) at (0,-3.4) {A supplied identification}; \node[draw,dashed,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (s) at (0,-5.2) {None of the candidates is supplied. The graph stays unresolved.}; \draw[->] (a) -- (b); \draw[->] (b) -- (c); \draw[->] (c) -- (s); \end{tikzpicture} \caption{Reading downward is not a transfer. The boxes name objects this manuscript keeps separate. The dashed box is the stated stop. Not a proof, and not an admission of a quantum coefficient.} \label{fig:chain-XV-E} \end{figure} \begin{figure}[ht] \centering \begin{tikzpicture}[x=1cm,y=1cm] \node[draw,rounded corners,align=center,text width=4.2cm,font=\small,inner sep=3pt] (a) at (0,0) {Source-recovery pass}; \node[align=center,font=\small] (m) at (5.2,0) {not the same object}; \node[draw,rounded corners,align=center,text width=4.2cm,font=\small,inner sep=3pt] (b) at (10.4,0) {A supplied identification}; \node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5.2,-2.35) {Four candidate symbols. None of the candidates is supplied. The graph stays unresolved.}; \draw (a) -- (m); \draw (m) -- (b); \draw[->] (m) -- (s); \end{tikzpicture} \caption{Source-recovery pass is not A supplied identification. The boxes name objects this manuscript keeps separate. The dashed box is the stated stop. Not a proof, and not an admission of a quantum coefficient.} \label{fig:pair-XV-E} \end{figure} \begin{figure}[ht] \centering \begin{tikzpicture}[x=1cm,y=1cm] \node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (a) at (0,0) {Named: Source-recovery pass}; \node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (b) at (5,0) {Used here: Four candidate symbols}; \node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (c) at (10,0) {Not claimed: A supplied identification}; \node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5,-2.5) {None of the candidates is supplied. The graph stays unresolved.}; \draw[->] (a) -- (b); \draw[->] (b) -- (c); \draw[->] (b) -- (s); \end{tikzpicture} \caption{Three statuses stay separate. The boxes name objects this manuscript keeps separate. The dashed box is the stated stop. Not a proof, and not an admission of a quantum coefficient.} \label{fig:gate-XV-E} \end{figure} Figure~\ref{fig:split-XV-E} records one view of the same separation. Figure~\ref{fig:chain-XV-E} records one view of the same separation. Figure~\ref{fig:pair-XV-E} records one view of the same separation. Figure~\ref{fig:gate-XV-E} records one view of the same separation. The drawings add no hypothesis. Figure~\ref{fig:scope-XV-E} records that separation. The drawing adds no hypothesis. \section{Purpose, scope, and outcomes} \label{sec:scope} \subsection{The question and the permitted endings} XV-D \cite{XVD} audited the supplied record item by item and found that the data defining the full joint characteristic operation for the frozen six-branch graph were not in it. XV-E asks: \begin{quote} Can the frozen six-branch operation be defined on the specified domain, extended to tests meeting the characteristic locus $N$, and made independent of presentation within the inherited correction class? \end{quote} Exactly one of three endings is permitted for the graph-level problem: \textbf{Constructed through descent}; \textbf{Obstructed in the frozen class}, with the class proved exhaustive; or \textbf{Unresolved at a named gate}. Failure to recover a source is not an analytic obstruction. Successful descent would not be quantum admission. \subsection{Gates and stop rule} \begin{center} \small \begin{tabularx}{\linewidth}{@{}p{0.07\linewidth}X@{}} \toprule Gate & Content\\ \midrule E0 & Source recovery: every item of XV-D's package (Definition D0.1--D0.20 of \cite{XVD}) needed to state the lift and its correction class is reproduced in the record and mutually consistent.\\ E1 & Definition of $\Fz:\Pint\to\Dp(W;E_{\mathrm{res}})$ on the actual presentation space with $r_O\Fz=m^\circ\Rec$ proved on the source-defined domain $O$.\\ E2 & Determination of $\ker\Rec$; evaluation of $d_0=\Fz|_{\ker\Rec}$; cancellation or noncancellation in the actual class $\mathcal A$.\\ E3 & Supported-sector estimates for the actual $v_\alpha$, $c_{D,\beta}$; characteristic compositions of $K_{47}$ and both $K_5$ placements.\\ E4 & Complete-graph estimates, output operations, admission.\\ \bottomrule \end{tabularx} \end{center} \textbf{E0 stop rule.} If an item needed to state the lift or its allowed corrections remains unavailable, this is said plainly; E1--E4 are not entered for the actual graph; no graph formula with placeholders is written and called a lift. \subsection{Evidence contract} Direct inputs: the full texts of XV-A, XV-B, XV-C, XV-D \cite{XVA,XVB,XVC,XVD}; the retrieved XIV-H, XIV-I, XIV-J excerpts \cite{Hkernel,Hverdict,Ipole,Iamplitude,Ijoint,Irelease,Jdescent, Jnull,Jreference,Jconditional,Jverdict,Jrelease,Jadmission}; the retrieved XIV-C and XIV-G release statements \cite{Crelease,Grelease}; the retrieved XII-A and XII-D passages \cite{XIIA,XIID}; and the proposed XV-E plan, used as a statement of objectives only. The coverage account \cite{record} is documentary context. The record itself states that similar words in distinct manuscripts do not identify their models, response maps, or sign conventions; this rule is applied to every candidate identification below. Nothing from XIV-A, XIV-B, XIII-D, or XIII-E is imported. \subsection{Numbering and admission} The technical-continuation convention of XV-B--XV-D is retained. The admission criterion \begin{equation} Q_H\Rquant\in\Eout \label{eq:admission} \end{equation} is a later project gate, not a prerequisite for this manuscript, and is not asserted here. \section{Source recovery (Gate E0)} \label{sec:recovery} \subsection{The recovery pass} The pass consists of a complete re-reading of the record as supplied with the present request, item by item against XV-D's Definition D0.1--D0.20. The record is the same retrieval corpus as before \cite{record}; the passages newly present in it, relative to the record audited by XV-D, are the XIV-C release \cite{Crelease}, the XII-A composite definitions \cite{XIIA}, and the XII-D root and transpose passage \cite{XIID}. No external registry is consulted, and no claim about documents outside the record is made. Status codes: \textsf{S} supplied (formula or definition reproduced); \textsf{T} transcribed from a reproduced excerpt; \textsf{C} candidate (a formula in the record for a differently named object, whose identification with the required symbol is not supplied); \textsf{R} reported only (name, locator, or sentence); \textsf{U} unavailable. \subsection{Manifest} \begin{center} \small \begin{tabularx}{\linewidth}{@{}p{0.06\linewidth} p{0.38\linewidth}p{0.07\linewidth}p{0.10\linewidth}X@{}} \toprule Item & Record content & Status & Needed for & Change since XV-D\\ \midrule D0.1 & $\Pint$: ``finite retarded-pole module of Definition def:amodule'' \cite{Iamplitude}. & \textsf{R} & E1, E2 & None.\\[0.25em] D0.2 & $\Rec$: two values, $\Rec(M_{\mathrm R})=\delta(q)R(r)$, $\Rec(i\theta M_{\mathrm R})=\delta'(q)R(r)$ \cite{Jnull}. & \textsf{T} (2 values) & E1, E2 & None.\\[0.25em] D0.3 & $\Kint=\im\Rec$ convention; topology unnamed. & \textsf{R} & E1 & None.\\[0.25em] D0.4 & $\ker\Rec$: one relation $(q\,i\theta+1)M_{\mathrm R}$ \cite{Jnull}. & \textsf{T} (1 relation) & E2 & None.\\[0.25em] D0.5 & $W$: name; variables $(a,b,z,y)$; $\dim W=16$ \cite{Grelease}. & \textsf{R} & E1 & None.\\[0.25em] D0.6 & $N=\Natt$: name; $\dim N=8$ \cite{Grelease}; a codimension-four stratum with rank-two cone differentials is named \cite{Crelease}. & \textsf{R} & E1 & XIV-C release present; no equations.\\[0.25em] D0.7 & $\Sresp$: name; codimension five; coordinates $q_1,q_2$ in \eqref{eq:referenceWF}. & \textsf{R} & E1, E3 & None.\\[0.25em] D0.8 & $O=\Ucan$: name; non-identity with $W\setminus\overline{\Natt}$ \cite{Ijoint}. & \textsf{R} & E1 & None.\\[0.25em] D0.9 & Collision strata, observable diagonals: names. & \textsf{U} & E4 & None.\\[0.25em] D0.10 & $m^\circ$: ``canonical grouped expression eq:target'' \cite{Ijoint}. & \textsf{R} & E1 & None.\\[0.25em] D0.11 & $\Cross$: name; traceability lemma lem:traces named \cite{Jreference}. & \textsf{R} & E1, E2 & None.\\[0.25em] D0.12 & Mixed vertex Hessian, weights: names. & \textsf{U} & E1 & None.\\[0.25em] D0.13 & $\mathsf D$ via undefined $\dot a_2$ \cite{Iamplitude}; pole chart of $R$ \eqref{eq:pole} \cite{Ipole}. Newly in record: $\mathfrak a^{\alpha\mu,ab}=\tfrac12\eta^{\alpha\mu}\eta^{ab} -\eta^{\alpha(a}\eta^{b)\mu}$, $\Lambda^{\mu,ab}(x,z)=\sigma_\eps(z)\mathfrak a^{\alpha\mu,ab} \partial^z_\alpha E_0^{\mathrm R}(x-z)$ \cite{XIIA}; $G^{\mathrm R}_{\mathrm{gr}}=-\mathfrak P E_0^{\mathrm R}$ \cite{XIID}. & \textsf{T}/\textsf{R}/\textsf{C} & E1, E3 & Candidates for $\mathsf A$, $L$, $R$, $\sigma_\eps$; see \S\ref{sec:candidates}.\\[0.25em] D0.14 & $K_8$, $K_9$: ``eq:K8reduced, eq:K9'' \cite{Hverdict}. & \textsf{R} & E1 & None.\\[0.25em] D0.15 & Normalization eq:Rconvention \cite{Jadmission}. & \textsf{R} & E4 & None.\\[0.25em] D0.16 & Candidate map class def:mapclass, def:class \cite{Jverdict,Ijoint}. & \textsf{R} & E2 & None.\\[0.25em] D0.17 & Correction class: ``XIII-C's partial-diagonal ambiguity class'' \cite{record}. & \textsf{R} & E2 & None.\\[0.25em] D0.18 & $v_\alpha$, $c_{D,\beta}$: symbols in \eqref{eq:reference} \cite{Jreference}. & \textsf{R} & E3 & None.\\[0.25em] D0.19 & Target cone: supported-sector bound \eqref{eq:referenceWF}; $\mathcal L(\cdot)$ named \cite{Jconditional}. & \textsf{T} (partial) & E4 & None.\\[0.25em] D0.20 & $Q_H$, $\Eout$, output operations: the line \eqref{eq:admission} \cite{Jadmission}. & \textsf{R} & E4 & None.\\ \bottomrule \end{tabularx} \end{center} \subsection{Candidate identifications} \label{sec:candidates} The XII-A passage \cite{XIIA} defines, on the XII sheet, \begin{equation} \mathfrak a^{\alpha\mu,ab} =\tfrac12\eta^{\alpha\mu}\eta^{ab}-\eta^{\alpha(a}\eta^{b)\mu}, \qquad \Lambda^{\mu,ab}(x,z) =\sigma_\eps(z)\,\mathfrak a^{\alpha\mu,ab}\, \partial^z_\alpha E_0^{\mathrm R}(x-z), \label{eq:XIIA} \end{equation} and the XII-D passage \cite{XIID} fixes the sign $G^{\mathrm R}_{\mathrm{gr}}=-\mathfrak P E_0^{\mathrm R}$ and displays $\alpha_{\mathrm{co}}^{ab} =-\sigma_\eps\mathfrak a^{\lambda\mu,ab}\partial_\lambda E_0^{\mathrm A}(f\partial_\mu\phi)$. The coverage account states that the XIV analytic branch inherits XII-C's cubic action and XII-A's carrier \cite{record}. The following identifications are therefore \emph{candidates}: \[ L^{\mu,J}(a,z)\ \overset{?}{=}\ \Lambda^{\mu,ab}(a,z)\ (J=(ab)), \qquad \mathsf A^{\alpha\beta,I}\ \overset{?}{=}\ \mathfrak a^{\alpha\beta,ab}\ (I=(ab)), \qquad R\ \overset{?}{=}\ E_0^{\mathrm R}, \qquad \sigma_\eps\ \overset{?}{=}\ \sigma_\eps . \] \begin{proposition}[Status of the candidates] \label{prop:candidates} None of the four identifications is supplied by the record. Each would be promoted to status \textsf{S} by reproducing XIV-H's equation \texttt{eq:L} and the definitions of $\mathsf A$ and $R$ in XIV-H or XIV-I, together with a statement in those manuscripts that their symbols are the XII-A objects, or by a transport proof meeting the four conditions of XV-A's domain-sensitive transfer criterion \cite{XVA}. Even if all four were promoted, items D0.1, D0.2, D0.4, D0.8, D0.10--D0.12, D0.14, D0.16--D0.18, and D0.20 would remain in status \textsf{R} or \textsf{U}. \end{proposition} \begin{proof} The XIV-H excerpt uses $L$ and refers to ``(eq:L)'' without reproducing it \cite{Hkernel}; the XIV-I excerpt uses $L$, $R$, $\mathsf D$ without definitions \cite{Iamplitude}. The XII-A passage names $\Lambda$ and $E_0^{\mathrm R}$ and does not mention XIV-H's symbols. The record's own reading rule forbids identifying objects across manuscripts by notation. The promotion conditions are read off from XV-A's transfer criterion. The last sentence is the manifest. \end{proof} \subsection{E0 verdict} \begin{proposition}[E0 fails] \label{prop:E0} The record does not define $\Pint$, $\Rec$ on $\Pint$, $O$, $m^\circ$, $\Cross$, the vertex data, the candidate class, or the correction class. Hence no preliminary lift $\Fz:\Pint\to\Dp(W;E_{\mathrm{res}})$ with $r_O\Fz=m^\circ\Rec$ can be stated, and no correction space $\mathcal A$ against which a null defect could be tested is available. \end{proposition} \begin{proof} A linear map requires a domain, a codomain, and a value on each element of the domain. The domain $\Pint$ (D0.1) and codomain (D0.5) are not defined. The value on any presentation of the six entries is the composition of the internal kernel with $\Cross$, the vertex Hessian, and weights (D0.10--D0.13), none of which is defined; the candidates of \S\ref{sec:candidates} concern only the internal kernels. The correction class needs D0.16--D0.17. All are in status \textsf{R}, \textsf{U}, or \textsf{C}. \end{proof} \[ \boxed{\text{E0: source recovery incomplete; D0 still fails.}} \] Gates E1--E4 are not entered for the actual graph. Section~\ref{sec:branches} records the freeze; Sections~\ref{sec:jet} and \ref{sec:erratum} contain the source-complete calculations. \section{Frozen ledger} \label{sec:branches} The retained kernel and supported sector are \begin{equation} \Kret=K_{47}+K_5+K_6+K_8+K_9,\qquad K_5=K_5^{[a]}+K_5^{[z]},\qquad \Ksup=K_6+K_8+K_9 . \label{eq:retained} \end{equation} $K_{47}$ groups $B_4+B_7$; both $K_5$ placements belong to $B_5$. The supplied internal expressions are \cite{Hkernel,Iamplitude} \begin{align} K_5^{[a],I,J}(a,b,z) &=\partial_{b^\mu}\Bigl[f(b)\sigma_\eps(a)\partial_{a^\alpha}R(b-a) \mathsf A^{\alpha\beta,I}\partial_{a^\beta}L^{\mu,J}(a,z)\Bigr], \label{eq:K5a}\\ K_5^{[z],I,J}(a,b,z) &=\partial_{b^\mu}\Bigl[f(b)\sigma_\eps(z)\partial_{z^\alpha}R(b-z) \mathsf A^{\alpha\beta,J}\partial_{z^\beta}L^{\mu,I}(z,a)\Bigr], \label{eq:K5z}\\ K_{47}^{I,J}(a,b,z) &=-\partial_{b^\mu}\Bigl[\bigl(f(b)\partial_{b^\nu}L^{\mu,J}(b,z) -\delta^\mu_\nu\partial_{b^\rho}(f(b)L^{\rho,J}(b,z))\bigr) L^{\nu,I}(b,a)\Bigr], \label{eq:K47}\\ K_6^{I,J}(a,b,z) &=\partial_{b^\mu}\Bigl[f(b)\sigma_\eps(a)\mathsf D^{\alpha\mu,I,J} \partial_{a^\alpha}R(b-a)\delta^{(4)}(a-z)\Bigr], \label{eq:K6} \end{align} and the internal pole chart is \begin{equation} M_{\mathrm R}=-\frac{s}{2\rho}(\omega-s\rho-i0)^{-1} +\frac{s}{2\rho(\omega+s\rho)},\qquad \rho>0,\ |\omega-s\rho|0$. There is $\theta_\eps\in C^\infty(\mathbb R^d)$ with $0\le\theta_\eps\le1$, $\theta_\eps=0$ on $\{\dist(x,Z)<\eps\}$, $\theta_\eps=1$ on $\{\dist(x,Z)>3\eps\}$, and $|\partial^\alpha\theta_\eps|\le C_\alpha\eps^{-|\alpha|}$ with $C_\alpha$ independent of $\eps$ and $Z$. \end{lemma} \begin{proof} Let $\rho\in\D(\mathbb R^d)$ be nonnegative, supported in the unit ball, with integral one, and $\rho_\eps(x)=\eps^{-d}\rho(x/\eps)$. Put $\theta_\eps=\mathbf 1_{\{\dist(\cdot,Z)\ge2\eps\}}*\rho_\eps$. It vanishes where $\dist<\eps$, equals one where $\dist>3\eps$, and $\partial^\alpha\theta_\eps=\mathbf 1*\partial^\alpha\rho_\eps$ is bounded by $\|\partial^\alpha\rho_\eps\|_{L^1}=\eps^{-|\alpha|} \|\partial^\alpha\rho\|_{L^1}$. \end{proof} \begin{lemma}[Taylor bound near $Z$] \label{lem:taylor} Let $K\subset X$ be compact, and let $\varrho>0$ be such that the closed $\varrho$-neighborhood $K'$ of $K$ is contained in $X$. Let $g\in C^\infty(X)$ have vanishing $k$-jet at every point of $Z$. Then for $x\in K$ with $\dist(x,Z)<\varrho$ and $|\alpha|\le k$, \[ |\partial^\alpha g(x)|\le C\,\dist(x,Z)^{\,k+1-|\alpha|}, \qquad C=\frac{d^{\,k+1}}{(k+1-|\alpha|)!} \max_{|\mu|=k+1}\sup_{K'}|\partial^\mu g| . \] \end{lemma} \begin{proof} Put $\delta=\dist(x,Z)<\varrho$ and choose $\delta<\varrho'<\varrho$. The closed ball $\bar B(x,\varrho')$ lies in $K'\subset X$, so $Z\cap\bar B(x,\varrho')$ is closed in $\bar B(x,\varrho')$, hence compact and nonempty; a nearest point $z\in Z$ with $|x-z|=\delta$ exists, and the segment $[z,x]\subset\bar B(x,\delta)\subset K'$. Taylor's formula for $\partial^\alpha g$ at $z$ to order $k-|\alpha|$, all of whose Taylor coefficients vanish, gives the integral-remainder bound stated, with the derivatives of order $k+1$ estimated on $K'$. \end{proof} \begin{lemma}[Jet-vanishing annihilators produce no defect] \label{lem:flat} Let $T$ extend $T^\circ$. Let $K$, $\varrho$, $K'$ be as in Lemma~\ref{lem:taylor}, and let $k\ge0$, $C_T$ be such that $|\langle T,\psi\rangle|\le C_Tp_k(\psi)$ for all $\psi\in\D(X)$ with $\supp\psi\subset K'$, where $p_k(\psi)=\max_{|\alpha|\le k}\|\partial^\alpha\psi\|_\infty$. If $g\in\Ann(T^\circ)$ has vanishing $k$-jet at every point of $Z$, then $\langle gT,\phi\rangle=0$ for every $\phi\in\D(X)$ with $\supp\phi\subset K$. Consequently, if $g\in\Ann(T^\circ)$ is flat on $Z$, then $gT=0$, i.e. $g\in\Ann(T)$. \end{lemma} \begin{proof} Put $\psi=g\phi$; its $k$-jet vanishes on $Z$ and $\supp\psi\subset K$. Let $\theta_\eps$ be as in Lemma~\ref{lem:cutoff} with $3\eps<\varrho$. Then $\theta_\eps\psi\in\D(X^\circ)$, so \[ \langle T,\theta_\eps\psi\rangle =\langle T^\circ,\theta_\eps g\phi\rangle =\langle g|_{X^\circ}T^\circ,\theta_\eps\phi\rangle=0 . \] Moreover $(1-\theta_\eps)\psi$ is supported in $K\cap\{\dist(\cdot,Z)\le3\eps\}$, and for $|\alpha|\le k$, by the Leibniz rule, Lemma~\ref{lem:cutoff}, and Lemma~\ref{lem:taylor} applied to $\psi$, \[ |\partial^\alpha((1-\theta_\eps)\psi)(x)| \le\sum_{\mu\le\alpha}\binom{\alpha}{\mu} C_\mu\eps^{-|\mu|}\,C\,(3\eps)^{k+1-|\alpha|+|\mu|} \le C'\eps^{\,k+1-|\alpha|}\le C'\eps . \] Hence $p_k((1-\theta_\eps)\psi)\to0$, and the order-$k$ estimate for $T$ on $K'$ gives $\langle T,\psi\rangle=\lim_{\eps\to0} \langle T,\theta_\eps\psi\rangle=0$. For flat $g$ every $k$ is admissible and every $\phi\in\D(X)$ has support in some compact $K$, so $gT=0$. \end{proof} \begin{theorem}[The defect is a finite-jet functional] \label{thm:jetdep} Let $T$ extend $T^\circ$, and let $K$, $\varrho$, $K'$, $k$ be as in Lemma~\ref{lem:flat}. For $a\in H$, the restriction of $d_0^T(a)$ to $\operatorname{int}(K)\times V_q$ depends only on the $k$-jets along $Z$ of the functions $\beta_\gamma(a)$, $|\gamma|\le r$. In particular: \begin{enumerate}[label=\textup{(\roman*)}] \item flat-annihilator relations carry no defect for any extension $T$; \item a nonzero defect is produced only by jet-detected relations. \end{enumerate} \end{theorem} \begin{proof} Let $a,a'\in H$ have $\beta_\gamma(a)-\beta_\gamma(a')$ with vanishing $k$-jet on $Z$ for all $\gamma$. Since $\beta_\gamma(a-a')\in\Ann(T^\circ)$, Lemma~\ref{lem:flat} gives $\langle\beta_\gamma(a-a')T,\psi\rangle=0$ for $\psi\in\D(X)$ with $\supp\psi\subset K$. For $\phi\in\D(U)$ supported in $\operatorname{int}(K)\times V_q$, each $\partial_q^\gamma\phi(\cdot,0)$ is such a $\psi$, so $\langle d_0^T(a)-d_0^T(a'),\phi\rangle =\sum_\gamma(-1)^{|\gamma|}\langle\beta_\gamma(a-a')T, \partial_q^\gamma\phi(\cdot,0)\rangle=0$. Taking $a'=0$ with a flat $a$ gives (i); (ii) is (i) together with XV-D's result on universal relations. \end{proof} \subsection{Point punctures: a descending extension always exists} \begin{theorem}[Descent by a finite delta correction] \label{thm:point} Let $Z=\{z_0\}$ and suppose $T^\circ$ has some extension $\tilde T\in\Dp(X)$. Let $\bar B=\bar B(z_0,\varrho_0)\subset X$ be a closed ball and $k$ an order of $\tilde T$ on $\bar B$. Then there is a distribution \[ s=\sum_{|\mu|\le k}c_\mu(-1)^{|\mu|}\partial^\mu\delta_{z_0}, \qquad c_\mu\in\mathbb C, \] such that $T=\tilde T-s$ extends $T^\circ$ and $\Ann(T)=\Ann(T^\circ)$. Consequently $H_T=H$, $d_0^T=0$, and $\Fz^T=\Fz^{\tilde T}-\Rec_s$ is a descending lift in the class of extension changes. \end{theorem} \begin{proof} Let $B=B(z_0,\varrho_0)$ be the open ball and $A_B=\{g\in\Ann(T^\circ):\supp g\subset B\}$. Define $\lambda:A_B\to\mathbb C$, $\lambda(g)=\langle\tilde T,g\rangle$. \emph{Step 1: $\lambda$ factors through $k$-jets.} Let $g\in A_B$ have vanishing $k$-jet at $z_0$. Choose $\chi\in\D(B)$ with $\chi=1$ on $\supp g$. Apply Lemma~\ref{lem:flat} with $K=\supp\chi$, $\varrho$ so small that $K'\subset\bar B$, and the order-$k$ bound of $\tilde T$ on $\bar B$: $\langle\tilde T,g\rangle =\langle g\tilde T,\chi\rangle=0$. Hence $\lambda(g)$ depends only on $j^k_{z_0}g=(\partial^\mu g(z_0))_{|\mu|\le k}$, i.e. $\lambda=\Lambda_0\circ j^k_{z_0}$ for a linear functional $\Lambda_0$ on the subspace $j^k_{z_0}(A_B)$ of the finite-dimensional jet space $J^k=\mathbb C^{\#\{|\mu|\le k\}}$. \emph{Step 2: the correction.} Extend $\Lambda_0$ linearly to $\Lambda$ on $J^k$ and write $\Lambda(j)=\sum_{|\mu|\le k}c_\mu j_\mu$. With $s$ as displayed, $\langle s,g\rangle=\sum_\mu c_\mu \partial^\mu g(z_0)=\Lambda(j^k_{z_0}g)$ for all $g\in\D(X)$. Set $T=\tilde T-s$. Since $\supp s=\{z_0\}\subset Z$, $T$ extends $T^\circ$. For $g\in A_B$, $\langle T,g\rangle=\lambda(g)-\Lambda(j^k_{z_0}g)=0$. \emph{Step 3: $\Ann(T)=\Ann(T^\circ)$.} The inclusion $\Ann(T)\subseteq\Ann(T^\circ)$ holds for every extension. Let $g\in\Ann(T^\circ)$ and $\phi\in\D(X)$; fix $\chi\in\D(B)$ with $\chi=1$ near $z_0$. Then $(1-\chi)g\phi\in\D(X^\circ)$ and $\langle T,(1-\chi)g\phi\rangle=\langle T^\circ,(1-\chi)g\phi\rangle =\langle g|_{X^\circ}T^\circ,(1-\chi)\phi\rangle=0$; and $\chi g\phi\in A_B$, so $\langle T,\chi g\phi\rangle=0$ by Step 2. Hence $\langle gT,\phi\rangle=0$ for all $\phi$, i.e. $g\in\Ann(T)$. The consequences follow from Theorem~\ref{thm:ann} and linearity of $T\mapsto\Rec_T$. \end{proof} \begin{remark} The theorem extends verbatim to finite $Z$, with one correction $s_i$ per point. For XV-D's example ($X=\mathbb R^2$, $Z=\{0\}$, $T^\circ=\delta(x_1)|_{X^\circ}$, $\tilde T=\delta(x_1)+\partial_1\delta_0$, $k=1$) one finds, for $g\in A_B$, $\lambda(g)=\int g(0,x_2)\dd x_2 -\partial_1g(0)=-\partial_1g(0)$, so $\Lambda(j)=-j_{e_1}$, $c_{e_1}=-1$, $s=\partial_1\delta_0$, and $T=\delta(x_1)$, in agreement with \cite{XVD}. \end{remark} \begin{corollary}[One dimension: every extension descends] \label{cor:1d} Let $d=1$, $Z=\{z_0\}$, and suppose $z_0\in\overline{\supp T^\circ}$ (closure in $X$). Then every $g\in\Ann(T^\circ)$ is flat at $z_0$, and consequently $\Ann(T)=\Ann(T^\circ)$ and $d_0^T=0$ for \emph{every} extension $T$ of $T^\circ$. \end{corollary} \begin{proof} If $g\in\Ann(T^\circ)$ and $g(p)\ne0$ for some $p\in\supp T^\circ$, then $g$ is invertible near $p$ and $T^\circ=g^{-1}(gT^\circ)=0$ near $p$, a contradiction; so $g=0$ on $\supp T^\circ$. By hypothesis there are points $x_n\in\supp T^\circ$, $x_n\to z_0$, $x_n\ne z_0$; passing to a subsequence on one side of $z_0$, $g$ has infinitely many zeros accumulating at $z_0$, so $g(z_0)=0$ and, by Rolle's theorem applied inductively, $g^{(j)}(z_0)=0$ for all $j$. Lemma~\ref{lem:flat} gives $gT=0$ for every extension $T$. \end{proof} \subsection{General punctures: the relative estimate} \begin{theorem}[Criterion for a descending extension] \label{thm:relative} Let $Z\subset X$ be relatively closed and $T^\circ\in\Dp(X^\circ)$. The following are equivalent. \begin{enumerate}[label=\textup{(\roman*)}] \item There is an extension $T\in\Dp(X)$ of $T^\circ$ with $\Ann(T)=\Ann(T^\circ)$ (equivalently, by Theorem~\ref{thm:ann}, $d_0^T=0$). \item For every compact $K\subset X$ there are $k_K\ge0$ and $C_K<\infty$ such that \begin{equation} |\langle T^\circ,\phi\rangle|\le C_K\,p_{k_K}(\phi+g) \label{eq:relest} \end{equation} for all $\phi\in\D(X^\circ)$ and $g\in\Ann(T^\circ)$ with $\supp\phi,\supp g\subset K$. \end{enumerate} \end{theorem} \begin{proof} (i)$\Rightarrow$(ii). Let $T$ be as in (i) and $K$ compact; let $k,C$ be an order and constant for $T$ on tests supported in $K$. For $\phi,g$ as in (ii), $\langle T,g\rangle=\langle gT,\chi\rangle=0$ with $\chi\in\D(X)$ equal to one on $\supp g$, so $|\langle T^\circ,\phi\rangle|=|\langle T,\phi+g\rangle-\langle T,g\rangle| =|\langle T,\phi+g\rangle|\le Cp_k(\phi+g)$. (ii)$\Rightarrow$(i). Fix a compact $K$ and let $\D_K=\{\psi\in\D(X):\supp\psi\subset K\}$, $S_K=\{\phi+g\}\subset\D_K$ with $\phi,g$ as in (ii). Define $\ell(\phi+g)=\langle T^\circ,\phi\rangle$. This is well defined: if $\phi+g=\phi'+g'$ then $h=\phi-\phi'=g'-g\in\D(X^\circ)\cap\Ann(T^\circ)$, and $\langle T^\circ,h\rangle=\langle hT^\circ,\chi\rangle=0$ for $\chi\in\D(X^\circ)$ equal to one on $\supp h$. By \eqref{eq:relest}, $|\ell|\le C_Kp_{k_K}$ on $S_K$. The Hahn--Banach theorem for functionals dominated by a seminorm extends $\ell$ to a linear $B_K$ on $\D_K$ with $|B_K|\le C_Kp_{k_K}$. Choose a locally finite cover of $X$ by relatively compact open sets $V_i$, compacts $K_i\supset\overline{V_i}$ in $X$, and a smooth partition of unity $\{\chi_i\}$ with $\supp\chi_i\subset V_i$. Define $\langle T,\phi\rangle=\sum_iB_{K_i}(\chi_i\phi)$ for $\phi\in\D(X)$; the sum is finite and $T$ is a distribution. If $\phi\in\D(X^\circ)$ then $\chi_i\phi\in\D(X^\circ)$, so $B_{K_i}(\chi_i\phi)=\ell(\chi_i\phi)=\langle T^\circ,\chi_i\phi\rangle$, and summing gives $T|_{X^\circ}=T^\circ$. If $g\in\Ann(T^\circ)$ and $\phi\in\D(X)$, then $\chi_ig\phi\in\Ann(T^\circ)$ with support in $K_i$, so $B_{K_i}(\chi_ig\phi)=\ell(0+\chi_ig\phi)=0$; hence $\langle gT,\phi\rangle=0$ and $g\in\Ann(T)$. Thus $\Ann(T^\circ)\subseteq\Ann(T)$, and the reverse inclusion holds for every extension. \end{proof} \begin{remark}[Scope of the criterion] \label{rem:open} Because $\phi+g$ coincides with $g$ near $Z$, $p_k(\phi+g)\ge\sup_Z\max_{|\alpha|\le k}|\partial^\alpha g|$, so \eqref{eq:relest} asks that $T^\circ$ be controlled by the $k$-jet of the annihilator $g$ along $Z$, tangential derivatives included. Theorem~\ref{thm:point} shows that for a point puncture \eqref{eq:relest} follows from the existence of any extension. For a puncture $Z$ that is not finite, the jet space along $Z$ is infinite-dimensional, and whether \eqref{eq:relest} can fail for an extendable $T^\circ$ is \emph{not decided in this release}. No example in either direction is asserted for such $Z$. \end{remark} \begin{corollary}[Relative obstruction in the jet module] \label{cor:relob} In the jet module with $Z$ finite, the relative obstruction class of any extension lift vanishes already in the class of extension changes $\{\Rec_s:\supp s\subset Z\}$, which is contained in XV-D's unrestricted puncture-supported class. For general $Z$ it vanishes in that class if and only if \eqref{eq:relest} holds. No separated contact witness exists for any extension lift, since every defect is supported in $Z\times\{0\}$ \cite{XVD}. \end{corollary} \begin{proof} Theorems~\ref{thm:point} and \ref{thm:relative}, together with $\Fz^{T}-\Fz^{\tilde T}=\Rec_{T-\tilde T}$ and XV-D's support theorem. \end{proof} None of the results of this section is transferred to $\Pint$: the record does not say whether $\ker\Rec$ contains annihilator-type relations, nor whether the actual target carries a finite normal-jet structure along a puncture. \section{Erratum to XV-D's coefficient proposition} \label{sec:erratum} XV-D, Section~6, Proposition \texttt{prop:reduction}, item (iii), states under hypothesis (H1) ($v_\alpha\in C^\infty(S_0)$) that if XV-C's estimate fails for $b^\circ_{D,\gamma}=v_\gamma c_{D,0}$, $|\gamma|=3$, then (a) $c_{D,0}$ has no extension to $\Dp(S_0)$, and (b) ``the failure sequence of XV-C for $b^\circ_{D,\gamma}$ is then a failure witness for the trace $c_{D,0}$ itself.'' Clause (a) is correct under (H1). Clause (b) overstates: the normalized form of XV-C's witness is not preserved under multiplication by $v_\gamma$. The proposed XV-E plan suggested instead that (a) requires an invertibility hypothesis; that suggestion conflates (a) with its converse. The following proposition settles all directions. \begin{proposition}[Extendability of a trace and of its smooth multiple] \label{prop:erratum} Let $S_0\subset\mathbb R^d$ be open, $Z\subset S_0$ relatively closed, $c\in\Dp(S_0\setminus Z)$, and let $p_k(\psi)=\max_{|\alpha|\le k}\|\partial^\alpha\psi\|_\infty$. \begin{enumerate}[label=\textup{(\alph*)}] \item If $v\in C^\infty(S_0)$ and $c$ extends to $\Dp(S_0)$, then $vc$ extends to $\Dp(S_0)$. Equivalently: if $vc$ has no extension, then $c$ has no extension. No invertibility of $v$ is needed. \item The converse of (a) fails: there are $v\in C^\infty(S_0)$ and $c$ without extension such that $vc$ extends. \item If $v$ is only assumed smooth on $S_0\setminus Z$, (a) fails: there are such $v$ and an extendable $c$ with $vc$ not extendable. \item (Corrected witness transfer.) Let $v\in C^\infty(S_0)$, let $K\subset S_0$ be compact, and let $(\phi_n)\subset\D(S_0\setminus Z)$, $\supp\phi_n\subset K$, satisfy $p_n(\phi_n)\le1$ and $|\langle vc,\phi_n\rangle|>n$. Put $\psi_n=v\phi_n$. Then $\supp\psi_n\subset K$, $|\langle c,\psi_n\rangle|>n$, and for each fixed $k$ there is $C_{k}=C_k(v,K)$ with $p_k(\psi_n)\le C_k$ for all $n\ge k$. Any sequence with these properties excludes an extension of $c$. However, $p_n(\psi_n)\le1$ need not hold, so $(\psi_n)$ is in general not a normalized failure sequence in the sense of XV-C; a normalized failure sequence for $c$ exists by XV-C's failure proposition but is in general a different sequence. \end{enumerate} \end{proposition} \begin{proof} (a) If $\tilde c\in\Dp(S_0)$ extends $c$, then $v\tilde c\in\Dp(S_0)$ and $(v\tilde c)|_{S_0\setminus Z}=v|_{S_0\setminus Z}c=vc$. (b) Take $S_0=(-1,1)$, $Z=\{0\}$, $c(x)=e^{1/x^2}$ for $x>0$ and $c(x)=0$ for $x<0$, which has no extension by XV-B's Proposition \texttt{prop:nonextendable} \cite{XVB}. Take $v(x)=e^{-2/x^2}$ for $x\ne0$, $v(0)=0$, which is in $C^\infty(S_0)$ and flat at $0$. Then $vc(x)=e^{-1/x^2}$ for $x>0$ and $0$ for $x<0$, a bounded continuous function on $(-1,1)\setminus\{0\}$, which extends to $\Dp(S_0)$ as the corresponding $L^\infty$ function. (c) Take $S_0$, $Z$ as in (b), $c=1$ (extendable), and $v(x)=e^{1/x^2}$ on $S_0\setminus Z$. Then $vc$ is XV-B's nonextendable function on $x>0$ (and $e^{1/x^2}$ on $x<0$, which does not restore extendability: XV-B's test functions are supported in $x>0$). (d) $\supp\psi_n\subset\supp\phi_n\subset K$ and $\langle c,\psi_n\rangle=\langle vc,\phi_n\rangle$. By the Leibniz rule, $p_k(v\phi_n)\le 2^k\max_{|\alpha|\le k}\sup_K|\partial^\alpha v| \cdot p_k(\phi_n)\le C_k$ for $n\ge k$, since $p_k(\phi_n)\le p_n(\phi_n)\le1$. If $c$ had an extension $\tilde c$, then on tests supported in $K$, $|\langle\tilde c,\psi\rangle|\le C p_k(\psi)$ for some $C,k$, so $n<|\langle c,\psi_n\rangle|\le CC_k$ for all $n\ge k$, a contradiction. For the last assertion, $p_n(v\phi_n)$ involves derivatives of $v$ of order up to $n$ on $K$, whose suprema are not bounded uniformly in $n$ in general; so the bound $p_n(\psi_n)\le1$ is not implied. Existence of a normalized sequence for $c$ is XV-C's Proposition \texttt{prop:failure} applied to $c$, which is nonextendable by (a) \cite{XVC}. \end{proof} \begin{corollary}[Replacement text for XV-D, Prop.\ prop:reduction (iii)] Under (H1)--(H2) of \cite{XVD}: if \eqref{eq:jetboundE} below fails for $b^\circ_{D,\gamma}=v_\gamma c_{D,0}$ with $|\gamma|=3$, then $c_{D,0}$ has no extension to $\Dp(S_0)$, and the sequence $v_\gamma\phi_n$ built from XV-C's failure sequence for $b^\circ_{D,\gamma}$ excludes an extension of $c_{D,0}$ in the sense of Proposition~\ref{prop:erratum}(d). Conversely, extendability of $b^\circ_{D,\gamma}$ does not imply extendability of $c_{D,0}$ without a further hypothesis on $v_\gamma$ near $Z$. \end{corollary} Here, as in XV-C and XV-D, \begin{equation} |\langle b^\circ_{D,\gamma},\psi\rangle|\le C_{K,\gamma}\, p_{k_{K,\gamma}}(\psi),\qquad \psi\in\D(S_0\setminus Z),\ \supp\psi\subset K , \label{eq:jetboundE} \end{equation} is the compact-uniform estimate equivalent to unrestricted supported realization of the normal form \eqref{eq:reference} \cite{XVC}. No instance of \eqref{eq:jetboundE} is verified or refuted for an actual coefficient, because no actual $v_\alpha$, $c_{D,\beta}$ is defined (D0.18). Gate E3 is not entered. \section{Correction class, contacts, and graph-level estimates} \label{sec:rest} For the actual graph, $\mathcal A$ is undefined (D0.16--D0.17), so neither $d_0$ nor $\Hom(\ker\Rec,Y)/\{a|_{\ker\Rec}\}$ is evaluable; XV-B's relative-lifting criterion cannot be instantiated. For the jet module, Corollary~\ref{cor:relob} records the exact situation. No obstruction and no cancellation is asserted for the graph. The complete-graph estimates require the equations of $W$, $N$, $\Sresp$, the collision strata, and the observable diagonals (D0.5--D0.9), of which the record supplies only dimensions and codimensions \cite{Grelease,Crelease,record}. No statement is made about a complete-graph wavefront bound, transverse estimates, contact compatibility, output operations, or \eqref{eq:admission}. Gate E4 is not entered. \section{Verdict and release} \label{sec:verdict} \begin{theorem}[XV-E local mathematical release] \label{thm:release} For the finite normal-jet module with punctured reconstruction of Section~\ref{sec:jet}: \begin{enumerate}[label=\textup{(\roman*)}] \item $H=\beta^{-1}(\Ann(T^\circ)^{M_r})$, $H_T=\beta^{-1}(\Ann(T)^{M_r})$, and $H/H_T\cong(\Ann(T^\circ)/\Ann(T))^{M_r}$; the null defect of $\Fz^T$ vanishes iff $\Ann(T)=\Ann(T^\circ)$; \item null relations are universal, flat-annihilator, or jet-detected; the first two carry no defect for any extension, and on a compact set the defect is a functional of the $k$-jets of $\beta_\gamma(a)$ along $Z$, $k$ an order of $T$ there; \item for a finite puncture, a descending extension exists whenever any extension exists, obtained by subtracting finitely many derivatives of the deltas at the puncture; in one dimension, if the puncture lies in the closure of $\supp T^\circ$, every extension descends; \item for a general puncture, a descending extension exists iff the relative estimate \eqref{eq:relest} holds; whether it can fail is not decided. \end{enumerate} In addition, Proposition~\ref{prop:erratum} corrects XV-D's Proposition \texttt{prop:reduction}(iii) as stated in Section~\ref{sec:erratum}. \end{theorem} \begin{proof} (i) is Theorem~\ref{thm:ann}; (ii) is Definition~\ref{def:kinds}, Lemma~\ref{lem:flat}, and Theorem~\ref{thm:jetdep}; (iii) is Theorem~\ref{thm:point} and Corollary~\ref{cor:1d}; (iv) is Theorem~\ref{thm:relative} and Remark~\ref{rem:open}. \end{proof} \subsection{Full-graph verdict} \begin{equation} \boxed{ \begin{gathered} \text{Unresolved at E0: source recovery incomplete; D0 still fails.}\\ \text{Four candidate identifications recorded, none supplied.}\\ \text{Gates E1--E4 not entered for the actual graph.} \end{gathered} } \label{eq:verdict} \end{equation} This is justified by Propositions~\ref{prop:candidates} and \ref{prop:E0}. It continues the J2, B1, C0, and D0 stops \cite{Jverdict,XVB,XVC,XVD}. It is not an obstruction and not a construction. The admission \eqref{eq:admission} is unasserted; ``not admitted'' is not ``nonexistent'' \cite{Jadmission}. \subsection{Acceptance ledger} \begin{center} \small \begin{tabularx}{\linewidth}{@{}p{0.38\linewidth}X@{}} \toprule Acceptance item & XV-E verdict\\ \midrule Source manifest complete for the claimed operation & No. One status change (D0.13 candidates); E0 fails.\\[0.25em] All six operational entries retained & Yes; freeze check passed.\\[0.25em] Preliminary lift defined on the claimed domain & No (Proposition~\ref{prop:E0}).\\[0.25em] Canonical agreement proved & Not applicable; no lift.\\[0.25em] Full reconstruction kernel handled & Graph: no. Jet module: complete, with classification (Theorem~\ref{thm:jetdep}).\\[0.25em] Defect computed on the kernel & Graph: no. Jet module: yes, with descending extension for finite punctures (Theorem~\ref{thm:point}).\\[0.25em] Cancellation in the actual class / exhaustive noncancellation & Class undefined; neither asserted.\\[0.25em] Coefficient estimates for actual formulas & Formulas unavailable; XV-D erratum resolved (Proposition~\ref{prop:erratum}).\\[0.25em] Complete-graph and output results & Not entered.\\[0.25em] Quantum admission & Unasserted.\\ \bottomrule \end{tabularx} \end{center} \section{Debugging record} \label{sec:debug} \begin{enumerate}[label=\textup{D\arabic*.},leftmargin=*] \item \textbf{Candidates are not definitions.} The XII-A and XII-D formulas are recorded under a separate status code \textsf{C}. They are not substituted into \eqref{eq:K5a}--\eqref{eq:K6}, and no calculation uses them. \item \textbf{Scope of the recovery pass.} The pass is over the supplied record only. No external search is reported as a source, and no statement about documents outside the record is made. \item \textbf{Erratum direction.} The plan's proposed erratum said that clause (a) of XV-D's item (iii) needs invertibility of $v_\gamma$. Under (H1) it does not (Proposition~\ref{prop:erratum}(a)); invertibility-type hypotheses are needed for the converse (Proposition~\ref{prop:erratum}(b)). The genuine overstatement in XV-D is the witness-transfer clause, corrected in (d). \item \textbf{Two annihilators.} $\Ann(T^\circ)$ is defined by annihilation on $X^\circ$ and $\Ann(T)$ on $X$; conflating them would make every defect vanish trivially. The isomorphism \eqref{eq:iso} depends on the surjectivity of $\beta$ (Lemma~\ref{lem:surj}). \item \textbf{Nearest points.} Lemma~\ref{lem:taylor} uses a nearest point of $Z$, which exists because $Z$ is relatively closed and the relevant closed ball lies in $X$; the hypothesis $\dist(x,Z)<\varrho$ is essential and is enforced in Lemma~\ref{lem:flat} by $3\eps<\varrho$. \item \textbf{Finite versus infinite punctures.} The descent theorem uses the finite dimension of the jet space at a point. For non-finite $Z$ the question is left open (Remark~\ref{rem:open}); no guess is recorded as a result. \item \textbf{Class of the correction.} The correction $s$ in Theorem~\ref{thm:point} is a change of extension, supported in $Z$; it is a statement about the jet module and is not offered as a graph prescription. \item \textbf{Branch counting.} Six operational entries; $K_{47}$ groups two originals; both $K_5$ placements belong to $B_5$; the supported sector is not the full sum. \item \textbf{Prescription.} Pole, propagators, carrier, normalization, and ambiguity class are unchanged. \item \textbf{Compilation.} The preambles of XV-A through XV-D use \verb|\mathscr| without loading a package that defines it; \texttt{mathrsfs} is loaded here. All macros used are defined; all \verb|\ref| and \verb|\cite| targets exist; \texttt{tabularx} column widths sum to less than \verb|\linewidth|; no non-ASCII characters occur in mathematics. \end{enumerate} \section{Successor requirement} \label{sec:successor} The next step requires the items D0.1, D0.2, D0.4, D0.8, D0.10--D0.12, D0.14, D0.16--D0.18, D0.20 in status \textsf{S}, and the promotion or rejection of the four candidates of \S\ref{sec:candidates} by reproduction of XIV-H's \texttt{eq:L} and the XIV-H/XIV-I definitions of $\mathsf A$ and $R$. With them, the programme of Section~\ref{sec:jet} is to be repeated for the actual module: define $\Fz$ on each presentation by the supplied composition rule; prove $r_O\Fz=m^\circ\Rec$ on the supplied domain; compute $\ker\Rec$ and classify its relations as universal, flat-annihilator, or jet-detected relative to the supplied characteristic locus; evaluate $d_0$ on the jet-detected relations; and decide cancellation in the supplied class $\mathcal A$. If the actual target has a finite normal-jet structure along a finite puncture, Theorem~\ref{thm:point} would apply; whether it does is not known from the record. The characteristic compositions of $K_{47}$ and both $K_5$ placements require a composition theorem not in the record and not supplied here. \subsection*{Final release sentence} \begin{center} \fbox{\parbox{0.91\linewidth}{ XV-E performs the source recovery pass over the supplied record, finds four candidate identifications and no new definitions, and therefore defines no lift for the actual graph and asserts no graph-specific defect, obstruction, estimate, or admission. For the punctured normal-jet module it proves the annihilator criterion for the null defect, the three-way classification of null relations, the finite-jet dependence of the defect, the existence of a descending extension for finite punctures, and the relative-estimate criterion for general punctures, and it corrects XV-D's coefficient proposition. The full joint characteristic map remains unresolved before instantiation. }} \end{center} \begin{thebibliography}{99} \bibitem{XVA} Manuscript XV-A, supplied \texttt{XV-A-va.tex}. \emph{A Model-and-Domain Audit of the Ramp, Graviton, and Joint-Extension Tracks}. Full text supplied. Used for the domain-sensitive transfer criterion. \bibitem{XVB} Manuscript XV-B, supplied \texttt{XV-B-va.tex}. \emph{Relative Lifting and Reconstruction Descent for the Full Joint Characteristic Target}. Full text supplied. Used for the linear interface, the relative-lifting criterion, the contact-support test, and Proposition \texttt{prop:nonextendable}. \bibitem{XVC} Manuscript XV-C, supplied \texttt{XV-C-va.tex}. \emph{The Full Joint Characteristic Target: Verified Reconstruction Tests and the Instantiation Boundary}. Full text supplied. Used for the extension criterion, Proposition \texttt{prop:failure}, and the supported realization theorem. \bibitem{XVD} Manuscript XV-D, supplied \texttt{XV-D-va.tex}. \emph{Instantiating the Frozen Joint Graph: Source Recovery, the Null Defect of Extension Lifts, and the Boundary of the First Full Calculation}. Full text supplied. Used for Definition D0.1--D0.20, the punctured normal-jet module, Theorems \texttt{thm:kernels} and \texttt{thm:defect}, Proposition \texttt{prop:example}, and Proposition \texttt{prop:reduction}, the last corrected here. \bibitem{Hkernel} Manuscript XIV-H, version~1. \texttt{XIV-H.tex}, identifier \texttt{05d90ea1}. Proposition \texttt{prop:K5exact}, lines 1135--1202. Direct excerpt supplied; \texttt{eq:L} is cited there and not reproduced. \bibitem{Hverdict} Manuscript XIV-H, version~1. Theorem \texttt{thm:H2}, lines 1539--1582; Theorem \texttt{thm:release}, lines 1847--1895. Direct excerpts supplied. \bibitem{Ipole} Manuscript XIV-I, version~1. \texttt{XIV-I.tex}, identifier \texttt{13365486}. Proposition \texttt{prop:pole}, lines 726--826. Direct excerpt supplied. \bibitem{Iamplitude} Manuscript XIV-I, version~1. Theorem \texttt{thm:I2} and adjacent precomposition formulas, lines 1288--1378. Direct excerpt supplied. \bibitem{Ijoint} Manuscript XIV-I, version~1. Proposition \texttt{prop:canonical}, Theorem \texttt{thm:I3}, lines 1459--1548. Direct excerpt supplied. \bibitem{Irelease} Manuscript XIV-I, version~1. Theorem \texttt{thm:release}, lines 1729--1814. Direct excerpt supplied. \bibitem{Jdescent} Manuscript XIV-J, version~1. \texttt{XIV-J.tex}, identifier \texttt{3c48e041}. Proposition \texttt{prop:descent}, lines 813--846. Direct excerpt supplied. \bibitem{Jnull} Manuscript XIV-J, version~1. Lemma \texttt{lem:null}, lines 853--883. Direct excerpt supplied. \bibitem{Jreference} Manuscript XIV-J, version~1. Theorem \texttt{thm:reference}, lines 1215--1299. Direct excerpt supplied; $v_\alpha$, $c_{D,\beta}$ not defined there. \bibitem{Jconditional} Manuscript XIV-J, version~1. Theorem \texttt{thm:conditionalextension}, lines 1427--1470. Direct excerpt supplied. \bibitem{Jverdict} Manuscript XIV-J, version~1. Theorem \texttt{thm:J2}, lines 1608--1649. Direct excerpt supplied. \bibitem{Jrelease} Manuscript XIV-J, version~1. Theorem \texttt{thm:release}, lines 1894--1939. Direct excerpt supplied. \bibitem{Jadmission} Manuscript XIV-J, version~1. Corollary \texttt{cor:noadmission}, lines 1808--1822, and admission-interface passage, lines 1794--1807. Direct excerpts supplied. \bibitem{Crelease} Manuscript XIV-C, version~1. \texttt{XIV-C.tex}, identifier \texttt{97aa7215}. Theorem \texttt{thm:release}, lines 1307--1340. Excerpt supplied in the retrieval account. Used only for the named codimension-four stratum. \bibitem{Grelease} Manuscript XIV-G, version~1. \texttt{XIV-G.tex}, identifier \texttt{4fcac266}. Theorem \texttt{thm:release}, lines 1710--1751. Excerpt supplied. Used only for the dimensions of $W$ and $N$. \bibitem{XIIA} Manuscript XII-A, version~2. \texttt{XII-A-v2.tex}, identifier \texttt{acb05c0f}. ``Named composites and their formal sector,'' lines 969--1059. Excerpt supplied. Source of the candidate formulas \eqref{eq:XIIA}; no identification with XIV-H symbols is supplied. \bibitem{XIID} Manuscript XII-D, version~1. \texttt{XII-D.tex}, identifier \texttt{e3e3f67d}. ``The first root and its transpose,'' lines 571--611. Excerpt supplied. Source of the sign relation $G^{\mathrm R}_{\mathrm{gr}}=-\mathfrak PE_0^{\mathrm R}$ on the XII sheet. \bibitem{record} Supplied coverage account, retrieval corpus identifier \texttt{3883b893}, retrieval~3. Explicitly disclaims a complete corpus scan and independent verification. Used only as documentary context and for reported codimension bookkeeping. \end{thebibliography} \end{document}